First, connect to the Time Capsule via an Ethernet cable if at all possible; it will be 2-3 times faster than via wireless.

If you're running 10.6.3 or later, you may be able to fix

the backups by holding the Alt/Option key while selecting the Time Machine icon in your menubar, and selecting the Verify Backups option.

Effective with Mountain Lion 10.8.x, if you're using multiple backup drives, this will verify the current or most recent backup drive only. To verify a different one, either run (or start) a backup to the desired destination, or use the procedure below.

If not, or if that doesn't work, or doesn't solve the problem, use the following procedure:

If Repair Disk fails:

If Disk Utility says it can't repair the sparse bundle, look closely at the messages. If it repaired some things, but not everything, run it again (and again) until it either repairs everything or can't repair any more.

If Disk Utility can't repair the sparse bundle, it's possible a heavy-duty 3rd-party disk repair app, such as DiskWarrior, can. These are expensive (DiskWarrior is about $100), and probably a good investment for the future, but there's no guarantee it will work. You must be able to "mount" the disk image by double-clicking the sparse bundle for these apps to be able to see and repair it.

If you decide to do this, be sure to use only a version that's compatible with the version of OSX on your system. (If you're not sure what version of OSX you're using, see What Version of OSX do I have?)

One Last Option:

If you're comfortable with Terminal and UNIX, the procedure in this post might be worth a try: http://blog.jthon.com/?p=31 It was written several years ago, and may or may not work in your situation, especially on later versions of OSX.

If you're out of options:

You might be able to recover some of the data, but it’s "iffy" and expensive. See Data Recovery for the gory details.

Otherwise, your only option is to erase the backups and let Time Machine start over.

If there's nothing else on the Time Capsule, erase it, per #Q5 in Using Time Machine with a Time Capsule. But doselect Zero Out Disk. That will take a long time, but if it fails, you know the TC's disk itself has failed. If it succeeds, the disk may be ok.

If there is other data on the Time Capsule, delete the sparse bundle via the Finder. That will take a long time, too. If it's extremely slow, you might be able to do it somewhat faster, per the blue box of #Q5 in Using Time Machine with a Time Capsule.

sparse bundle, the messages may not appear in the center area of the screen. Click the Log icon in the toolbar, or select Window > Show Log from the Menubar to display them.

What you're looking for, of course, is the "appears to be ok" message. If you get error messages instead, see the pinkbox below.

Note that this log is kept on your Mac, so if you missed the messages, you can start Disk Utility again and go directly to them, without having to run Repair Disk again. Each task is separated by a row of asterisks.

Backups made over a network are stored in a container called a sparse bundle disk image.

Disk Utility can't see the Time Capsule's internal HD directly, as it can an internal or external HD, but it can see the sparse bundle. If thesparse bundle containing your backups does not appear in Disk Utility's sidebar

(like the second sample below), open a Finder window, double-click the Time Capsule in the sidebar, then click the Data folder (it may have a different name) and locate the sparsebundle (the sample is the Finder in Column View).

(Note: The .sparsebundle suffix isn't always displayed.)

If the server isn't shown, from a Finder menubar, select Preferences > Sidebar and make sure the Connected Servers box is checked.

If you're backing-up more than one Mac to the Time Capsule, there will be a separate sparse bundle for each Mac. Drag the one you want from the Finder window to the lower part of Disk Utility's sidebar. It may take a moment to appear.

Select the sparse

bundle in the sidebar, click the First Aid tab near the top, then the Repair Disk button.